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<div class="section">
<div class="titlepage"><div><div><h3 class="title">
<a name="math_toolkit.internals.minimax"></a><a class="link" href="minimax.html" title="Minimax Approximations and the Remez Algorithm">Minimax Approximations
      and the Remez Algorithm</a>
</h3></div></div></div>
<p>
        The directory <code class="computeroutput"><span class="identifier">libs</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">minimax</span></code>
        contains an interactive command-line driven program for the generation of
        minimax approximations using the Remez algorithm. Both polynomial and rational
        approximations are supported, although the latter are tricky to converge:
        it is not uncommon for convergence of rational forms to fail. No such limitations
        are present for polynomial approximations which should always converge smoothly.
      </p>
<p>
        It's worth stressing that developing rational approximations to functions
        is often not an easy task, and one to which many books have been devoted.
        To use this tool, you will need to have a reasonable grasp of what the Remez
        algorithm is, and the general form of the approximation you want to achieve.
      </p>
<p>
        Unless you already familiar with the Remez method, you should first read
        the <a class="link" href="../remez.html" title="The Remez Method">brief background article explaining
        the principles behind the Remez algorithm</a>.
      </p>
<p>
        The program consists of two parts:
      </p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">main.cpp</span></dt>
<dd><p>
              Contains the command line parser, and all the calls to the Remez code.
            </p></dd>
<dt><span class="term">f.cpp</span></dt>
<dd><p>
              Contains the function to approximate.
            </p></dd>
</dl>
</div>
<p>
        Therefore to use this tool, you must modify f.cpp to return the function
        to approximate. The tools supports multiple function approximations within
        the same compiled program: each as a separate variant:
      </p>
<pre class="programlisting"><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span> <span class="identifier">f</span><span class="special">(</span><span class="keyword">const</span> <span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span><span class="special">&amp;</span> <span class="identifier">x</span><span class="special">,</span> <span class="keyword">int</span> <span class="identifier">variant</span><span class="special">);</span>
</pre>
<p>
        Returns the value of the function <span class="emphasis"><em>variant</em></span> at point
        <span class="emphasis"><em>x</em></span>. So if you wish you can just add the function to approximate
        as a new variant after the existing examples.
      </p>
<p>
        In addition to those two files, the program needs to be linked to a <a class="link" href="../high_precision/use_ntl.html" title="Using NTL Library">patched NTL library to compile</a>.
      </p>
<p>
        Note that the function <span class="emphasis"><em>f</em></span> must return the rational part
        of the approximation: for example if you are approximating a function <span class="emphasis"><em>f(x)</em></span>
        then it is quite common to use:
      </p>
<div class="blockquote"><blockquote class="blockquote"><p>
          <span class="serif_italic">f(x) = g(x)(Y + R(x))</span>
        </p></blockquote></div>
<p>
        where <span class="emphasis"><em>g(x)</em></span> is the dominant part of <span class="emphasis"><em>f(x)</em></span>,
        <span class="emphasis"><em>Y</em></span> is some constant, and <span class="emphasis"><em>R(x)</em></span> is
        the rational approximation part, usually optimised for a low absolute error
        compared to |Y|.
      </p>
<p>
        In this case you would define <span class="emphasis"><em>f</em></span> to return <span class="serif-italic">f(x)/g(x)</span>
        and then set the y-offset of the approximation to <span class="emphasis"><em>Y</em></span>
        (see command line options below).
      </p>
<p>
        Many other forms are possible, but in all cases the objective is to split
        <span class="emphasis"><em>f(x)</em></span> into a dominant part that you can evaluate easily
        using standard math functions, and a smooth and slowly changing rational
        approximation part. Refer to your favourite textbook for more examples.
      </p>
<p>
        Command line options for the program are as follows:
      </p>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term">variant N</span></dt>
<dd><p>
              Sets the current function variant to N. This allows multiple functions
              that are to be approximated to be compiled into the same executable.
              Defaults to 0.
            </p></dd>
<dt><span class="term">range a b</span></dt>
<dd><p>
              Sets the domain for the approximation to the range [a,b], defaults
              to [0,1].
            </p></dd>
<dt><span class="term">relative</span></dt>
<dd><p>
              Sets the Remez code to optimise for relative error. This is the default
              at program startup. Note that relative error can only be used if f(x)
              has no roots over the range being optimised.
            </p></dd>
<dt><span class="term">absolute</span></dt>
<dd><p>
              Sets the Remez code to optimise for absolute error.
            </p></dd>
<dt><span class="term">pin [true|false]</span></dt>
<dd><p>
              "Pins" the code so that the rational approximation passes
              through the origin. Obviously only set this to <span class="emphasis"><em>true</em></span>
              if R(0) must be zero. This is typically used when trying to preserve
              a root at [0,0] while also optimising for relative error.
            </p></dd>
<dt><span class="term">order N D</span></dt>
<dd><p>
              Sets the order of the approximation to <span class="emphasis"><em>N</em></span> in the
              numerator and <span class="emphasis"><em>D</em></span> in the denominator. If <span class="emphasis"><em>D</em></span>
              is zero then the result will be a polynomial approximation. There will
              be N+D+2 coefficients in total, the first coefficient of the numerator
              is zero if <span class="emphasis"><em>pin</em></span> was set to true, and the first
              coefficient of the denominator is always one.
            </p></dd>
<dt><span class="term">working-precision N</span></dt>
<dd><p>
              Sets the working precision of NTL::RR to <span class="emphasis"><em>N</em></span> binary
              digits. Defaults to 250.
            </p></dd>
<dt><span class="term">target-precision N</span></dt>
<dd><p>
              Sets the precision of printed output to <span class="emphasis"><em>N</em></span> binary
              digits: set to the same number of digits as the type that will be used
              to evaluate the approximation. Defaults to 53 (for double precision).
            </p></dd>
<dt><span class="term">skew val</span></dt>
<dd><p>
              "Skews" the initial interpolated control points towards one
              end or the other of the range. Positive values skew the initial control
              points towards the left hand side of the range, and negative values
              towards the right hand side. If an approximation won't converge (a
              common situation) try adjusting the skew parameter until the first
              step yields the smallest possible error. <span class="emphasis"><em>val</em></span> should
              be in the range [-100,+100], the default is zero.
            </p></dd>
<dt><span class="term">brake val</span></dt>
<dd><p>
              Sets a brake on each step so that the change in the control points
              is braked by <span class="emphasis"><em>val%</em></span>. Defaults to 50, try a higher
              value if an approximation won't converge, or a lower value to get speedier
              convergence.
            </p></dd>
<dt><span class="term">x-offset val</span></dt>
<dd><p>
              Sets the x-offset to <span class="emphasis"><em>val</em></span>: the approximation will
              be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
              where <span class="emphasis"><em>X</em></span> is the x-offset, <span class="emphasis"><em>S</em></span>
              is the x-scale and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
              to zero. To avoid rounding errors, take care to specify a value that
              can be exactly represented as a floating point number.
            </p></dd>
<dt><span class="term">x-scale val</span></dt>
<dd><p>
              Sets the x-scale to <span class="emphasis"><em>val</em></span>: the approximation will
              be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
              where <span class="emphasis"><em>S</em></span> is the x-scale, <span class="emphasis"><em>X</em></span>
              is the x-offset and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
              to one. To avoid rounding errors, take care to specify a value that
              can be exactly represented as a floating point number.
            </p></dd>
<dt><span class="term">y-offset val</span></dt>
<dd><p>
              Sets the y-offset to <span class="emphasis"><em>val</em></span>: the approximation will
              be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">S</span> <span class="special">*</span> <span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">))</span> <span class="special">+</span> <span class="identifier">Y</span></code>
              where <span class="emphasis"><em>X</em></span> is the x-offset, <span class="emphasis"><em>S</em></span>
              is the x-scale and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults
              to zero. To avoid rounding errors, take care to specify a value that
              can be exactly represented as a floating point number.
            </p></dd>
<dt><span class="term">y-offset auto</span></dt>
<dd><p>
              Sets the y-offset to the average value of f(x) evaluated at the two
              endpoints of the range plus the midpoint of the range. The calculated
              value is deliberately truncated to <span class="emphasis"><em>float</em></span> precision
              (and should be stored as a <span class="emphasis"><em>float</em></span> in your code).
              The approximation will be generated for <code class="computeroutput"><span class="identifier">f</span><span class="special">(</span><span class="identifier">x</span> <span class="special">+</span> <span class="identifier">X</span><span class="special">)</span> <span class="special">+</span> <span class="identifier">Y</span></code> where <span class="emphasis"><em>X</em></span> is
              the x-offset and <span class="emphasis"><em>Y</em></span> is the y-offset. Defaults to
              zero.
            </p></dd>
<dt><span class="term">graph N</span></dt>
<dd><p>
              Prints N evaluations of f(x) at evenly spaced points over the range
              being optimised. If unspecified then <span class="emphasis"><em>N</em></span> defaults
              to 3. Use to check that f(x) is indeed smooth over the range of interest.
            </p></dd>
<dt><span class="term">step N</span></dt>
<dd><p>
              Performs <span class="emphasis"><em>N</em></span> steps, or one step if <span class="emphasis"><em>N</em></span>
              is unspecified. After each step prints: the peek error at the extrema
              of the error function of the approximation, the theoretical error term
              solved for on the last step, and the maximum relative change in the
              location of the Chebyshev control points. The approximation is converged
              on the minimax solution when the two error terms are (approximately)
              equal, and the change in the control points has decreased to a suitably
              small value.
            </p></dd>
<dt><span class="term">test [float|double|long]</span></dt>
<dd><p>
              Tests the current approximation at float, double, or long double precision.
              Useful to check for rounding errors in evaluating the approximation
              at fixed precision. Tests are conducted at the extrema of the error
              function of the approximation, and at the zeros of the error function.
            </p></dd>
<dt><span class="term">test [float|double|long] N</span></dt>
<dd><p>
              Tests the current approximation at float, double, or long double precision.
              Useful to check for rounding errors in evaluating the approximation
              at fixed precision. Tests are conducted at N evenly spaced points over
              the range of the approximation. If none of [float|double|long] are
              specified then tests using NTL::RR, this can be used to obtain the
              error function of the approximation.
            </p></dd>
<dt><span class="term">rescale a b</span></dt>
<dd><p>
              Takes the current Chebeshev control points, and rescales them over
              a new interval [a,b]. Sometimes this can be used to obtain starting
              control points for an approximation that can not otherwise be converged.
            </p></dd>
<dt><span class="term">rotate</span></dt>
<dd><p>
              Moves one term from the numerator to the denominator, but keeps the
              Chebyshev control points the same. Sometimes this can be used to obtain
              starting control points for an approximation that can not otherwise
              be converged.
            </p></dd>
<dt><span class="term">info</span></dt>
<dd><p>
              Prints out the current approximation: the location of the zeros of
              the error function, the location of the Chebyshev control points, the
              x and y offsets, and of course the coefficients of the polynomials.
            </p></dd>
</dl>
</div>
</div>
<div class="copyright-footer">Copyright © 2006-2021 Nikhar Agrawal, Anton Bikineev, Matthew Borland,
      Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert Holin, Bruno
      Lalande, John Maddock, Evan Miller, Jeremy Murphy, Matthew Pulver, Johan Råde,
      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
      </p>
</div>
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